HALL PROBE GRADIOMETRY · 2019. 10. 14. · [HALL PROBE GRADIOMETRY ] A novel Scanning Hall probe gradiometer has been developed and a new method to image x, y & z components of the

2008

[HALL PROBE GRADIOMETRY ]

A novel Scanning Hall probe gradiometer has been developed and a new method to image x, y & z components of the magnetic field on the sample surface has been demonstrated for the first time with 1μm spatial resolution. 3D field distribution of a Hard Disk sample is successfully measured at 77K using this novel approach to prove the concept.

www.nanomagnetics‐inst.com 2

1.1 Novel Hall Probe Gradiometry

The gradiometry in the general sense describes the method used to measure the

variation of a physical quantity in space. In our case it is the magnetic field variation

over the sample. From Figure 1 it can be seen that if a different voltage value is

measured through terminals V1-2 and V3-4 due to the presence of an external magnetic

field up on the Hall array, this indicates a field variation along the current axis. The

magnitude of the voltage difference can be used to quantify this variation. This is a one

dimensional first order gradiometer configuration. Number of terminals can be

increased but the separation between them should be as much as the width of the Hall

crosses.

Figure 1: Hall gradiometer.

The method used in this study, however, is completely different than the methods

described above. There is no need for the complex design principles or the growth

requirements. Any semiconducting material suitable for a Hall effect device is

applicable with a simple fabrication process. The spatial resolution is just defined by the

width of the Hall cross, thus the limitation only comes from the fabrication capability or

from the material properties, such as the surface charge depletion after certain size. In

fact, probes with ~50nm spatial resolution has already been shown [1], which can

readily be employed to measure the three dimensional magnetic fields using that

method.

2w

w IH

V1

V2

V3

V4


The low noise amplifier is modified such that, the current and voltage signals can be

passed through any four leads of the Hall cross. The restriction of having the current and

the voltage leads to be mutually perpendicular to each other is eliminated. If the current

flow does not follow a straight path as shown in Fig. 2, we still observe the Hall effect,

which we may call the bending resistance [2-3]. As the current flows at any particular

point along the path, the magnetic force FM is perpendicular to the direction of motion.

In uniform magnetic field however, the Hall voltage would be equal to zero. But, if the

magnetic field distribution across the device is spatially non-uniform and

nonsymmetrical along the gradient axis, there will be a finite voltage developed at the

voltage terminals. The lateral electric field generated to balance the magnetic force is

not uniform across the device; in other words, it has a gradient along the device. As a

result, the measured potential is the derivative of the perpendicular magnetic field in the

active region of the Hall cross. Depending on the current flow path the derivative about

to different diagonals of the Hall cross is measured as shown in Fig. 3 with the non-

perpendicular current and voltage leads with different configurations and the field

gradient axis. This has been accidentally observed by Bath University group [4] during

an SHPM experiment, where the voltage and current leads were mixed up. The

orientation of the sensor over the magnetic field distribution of the measured sample is

important. In our setup the probe has a fixed position with respect to the quadrants of

the scanner piezo. Thus, the orientation of the probe is not changing during the scan

which is placed by 45º with respect to the X&Y scan directions (Fig.4). On the other

hand the sample can be aligned in any desired rotational angle with respect to our Hall

sensor. In the experiments a hard disk sample is imaged with a 1µm size PHEMT Hall

probe in LN2 environment to minimize the thermal drift. First, an image is obtained

using the normal Hall sensor configuration, where current and voltage leads were

perpendicular to each other.


Figure 2: Hall effect with non-perpendicular current and voltage leads with different

configurations under uniform magnetic field.

1

4

2

3

FL

B

IH

VG=0

.

‐‐

‐

‐‐

1

2

4

3

FL

B

IH

VG=0

.

‐‐

‐

‐‐

‐


Figure 3: Hall effect with non-perpendicular current and voltage leads with different

configurations under non-uniform magnetic field.

1

2

4

3

FL

BIH

VG=∂Bz(x,y)/∂y

.+

+

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x B

+

‐

4

2

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Gradient Axis

Gradient Axis


The scan performed with a speed of 5µm/s at 256 pixel resolution, applying a 500µA

DC drive current to the probe in the AFM tracking mode. Figure 5 shows the SHPM

image of the Hard Disk sample resolving individual bits. The Hall sensor alignment and

the lead configurations are given at the right hand side of the figure.

Figure 4: Orientation of the Hall sensor, over the sample, with respect to the scan

direction of the microscope: Hall cross has 45° alignment.

N

S

W E

Y scan direction

X scan direction


Figure 5: SHPM scan of a Hard disk sample at 77K with normal Hall sensor

configuration of mutually perpendicular current and voltage leads. Hall cross diagram

shows the relative alignment of the probe over the sample and the leads’ positions.

The magnitude of the Hall voltage signal is stored in a 256×256 matrix. Hence, matrix

arithmetic operations can be applied on the acquired image. Equipped with these, the

rows are differentiated along the y direction to obtain ∂Bz/∂x, and columns are

differentiated to get ∂Bz/∂y along the x direction using forward differences method. The

results are shown in Fig 6 and Fig 7 which are calculated ∂Bz/∂x and ∂Bz/∂y

respectively. Immediately after obtaining the standard SHPM image, drive current

direction is changed to the proper leads so that the ∂Bz/∂x and ∂Bz/∂y images can be

obtained. To have a healthy comparison, the probe was not pulled back off the surface

while the switch positions on the amplifier are changed. The Hall probe was driven by

smaller, 100µA current, as it was not possible to properly offset null the probe. The

other scan parameters are held the same as in the case of normal scan. Images obtained

as a result of these experiments and the current/ voltage lead configurations are shown

in Fig. 8 and 9. The results are in perfect agreement with the calculated ones, except the

field values are not absolute since the probe is offset nulled at the presence of the field.

I+V+

V‐I‐


Figure 6: Calculated ∂Bz/∂x from the measured Bz(x,y) data matrix by forward

differences.

Figure 7: Calculated ∂Bz/∂y from the measured Bz(x,y) data matrix by forward

differences.


Figure 8: SHPM scan of a HDD at 77K with Hall sensor configuration shown in the

diagram. The image represents ∂Bz/∂x due to the relative positions of current and

voltage leads

Figure 9: SHPM scan of a HDD at 77K with Hall sensor configuration shown in the

diagram. The image represents ∂Bz/∂y due to the relative positions of current and

voltage leads.

V+ı+

V‐I‐

I+I‐

V+V‐


1.2 Three Dimensional Magnetic Field Imaging

As different sample orientations are scanned, different images are obtained in lateral

derivative, which was actually expected as the gradient of the field changes in the space.

One may argue that since we can calculate the local gradient of the Bz(x,y) along x and

y directions, why should we bother imaging in this obscure way? On the other hand this

measurement mode clearly gives us the local gradient of the Bz(x,y), which can be used

in a number of applications like gradiometry etc.

A more important ramification is the possibility of calculating in-plane components,

Bx(x,y) and By(x,y) of the magnetic field if we measure Bz(x,y), ∂Bz(x,y)/∂x and

∂Bz(x,y)/∂y across the space. If we start with the Maxwell equation derived from

Ampere’s law [5],

B ε µ J ε µ E (1)

In a source free region the equation simplifies to,

B 0 (2) which can be written, in open form, as

̂ ̂ B ̂ B ̂ B 0 (3)

from which we can get,

B B ̂ B B ̂ B B 0 (4)

To satisfy the equation each vector component must individually be equal to zero.


B B 0 (5) B B 0 (6) B B 0 (7)

The first two equations can be solved in terms of the parameters we measure at the

beginning of the problem. Hence we can write,

B B 0 (8)

B B (9) B B dz (10)

Hence if we measure or obtain ∂Bz/∂y as a function of z, then we can calculate the By at

this specific z. The SHPM data should be obtained at increasing sample-sensor

distances, until the signal decays to zero or below the noise levels. A similar equation

can be written for Bx as well,

B x, y, z B , , dz (11)

As, it is not possible to integrate the equations analytically, we have to compute them

numerically from the measured data.

z dz ∑ (12)


The situation can visualized with the aid of Fig. 10 given below. While we can measure

the quantitative value of the perpendicular magnetic field, Bz, at the closest sample-

sensor position, this is not exactly the case for the derivatives. The change in magnitude

at a particular point for, ∂Bz(x,y,z)/∂x and ∂Bz(x,y,z)/∂y is not necessarily a decrease

with increasing sample-sensor separation. However, the overall dynamic field will

decrease with increasing probe-sample separation.

Note that we do not need to have an infinite sum as the field value f(jh) will decay after

a certain value of jh that can be set as the finite upper limit n. Thus, ∂Bz(x,y,z)/∂x and

∂Bz(x,y,z)/∂y are acquired at different heights (zi) until the field decays to zero,

carefully recording the values. A fixed incremental separation h is used between the

scans for ease of calculation.

Figure 10: Visualization of incremental scan for Bz(x,y,z), ∂Bz(x,y,z)/∂x and

∂Bz(x,y,z)/∂y

While the sample is scanned the surface is never perfectly smooth. In addition to this,

some inclination angle can unintentionally be given, while mounting the sample. For

this reason, while the sample is scanned with a liftoff, this distance must be applied

throughout the sample evenly. The magnetic force microscopy (MFM) techniques,

where the magnetic and interaction forces have to be separated, can be applied to this

h1

h0

h2

h3

hn z

x

y


case. Hence, while the forward scan is conducted using the feedback following the

surface texture; the backward scan is performed following the same texture, however

this time, by adding the required lift-off value at each pixels. By this procedure, the

value of h is held fixed throughout the whole scan area. Fig 11 shows the image of Hard

disk with the forward scan, (a) while the probe is in closest proximity of the sample, and

backward scan (b) while the probe is lifted off by 3.5µm. The decay of the magnetic

signal when the probe is lifted off by 3.5µm away from the surface can clearly be seen

and obviously this distance can be accepted as the upper limit of the integration while

calculating the Bx and By. This value is within the range of retraction capability of the

used scanner piezo at room temperature. But, unfortunately, the effect of the thermal

drift is also high at room temperature and it is virtually impossible to scan the same

portion of the HDD by giving successive offsets, which is required to integrate ∂Bz/∂x

or ∂Bz/∂y. Fig 12 shows this effect. At each level of 0.5µm offset, three images, Bz,

∂Bz/∂x and ∂Bz/∂y, had to be recorded. Each image requires ~40 minutes to be scanned.

Hence, approximately two hours spent at each level. As a result, the experiments had to

be done at low temperature to eliminate the thermal drift, which unfortunately reduces

the scan area of the piezo tube in X, Y and Z directions. To satisfy both requirements,

the stick-slip coarse approach mechanism is used to successively pull the Hall sensor off

the sample. The motion of the puck is first calibrated for a given voltage pulse. Then

appropriate numbers of pulses were applied in order to lift-off the sensor from the

sample. We adjusted the voltage pulses applied to the piezo to provide 250nm backward

steps. Overall 26 scans are performed, starting from the closest proximity of the sample,

with 250nm steps, until the probe is 6.5µm away from the sample in height. At each

height level, Bz(x,y), ∂Bz(x,y)/∂x and ∂Bz(x,y)/∂y were imaged. The thermal drift is

insignificant at 77K as evident in Fig 13. ∂Bz/∂x and ∂Bz/∂y images are used to integrate

Bx and By respectively using the method described above. Figures 14-16 show the Bz,

By and Bx fields of an Hard disk sample.

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Figure 14: SHPM image of Hard disk sample, shows the Bz of the field (a), obtained at

77K in the feedback tracking zone. Image was obtained using a 1µm PHEMT sensor.

Scan speed was 5µm/s, resolution set to 256×256 pixels. ∂Bz(x,y)/∂x (b) and ∂Bz(x,y)/∂y

(c) are calculated from Bz image by differentiating rows and columns of the image

matrix respectively.


Figure 15: By field calculated by integrating ∂Bz/∂y over a finite range. Each image file used in calculation was obtained using the same 1µm PHEMT sensor. Scan speed

was 5µm/s, resolution set to 256×256 pixels. The increments along the z-direction, h, is

set to 0.25μm in the range of [0, 6.5μm].


Figure 16: Bx field calculated by integrating ∂Bz/∂x over a finite range. Each image file

used in calculation was obtained using the same 1µm PHEMT sensor. Scan speed was

5µm/s, resolution set to 256×256 pixels. The increments along the z-direction, h, is set

to 0.25μm in the range of [0, 6.5μm].


1. Sandhu, A., et al., 50 nm Hall sensors for room temperature scanning Hall probe microscopy. Japanese Journal of Applied Physics Part 1-Regular Papers Short Notes & Review Papers, 2004. 43(2): p. 777-778.

2. Grundler, D., et al., Bend-resistance nanomagnetometry: spatially resolved magnetization studies in a ferromagnet/semiconductor hybrid structure. Physica E-Low-Dimensional Systems & Nanostructures, 2002. 12(1-4): p. 248-251.

3. Peeters, F.M. and X.Q. Li, Hall magnetometer in the ballistic regime. Applied Physics Letters, 1998. 72(5): p. 572-574.

4. Oral, A., unpublished. 5. Jackson, J.D., Classical Electrodynamics. 3rd ed. 1999, New York: Wiley. xxi,

808 p.

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HALL PROBE GRADIOMETRY · 2019. 10. 14. · [HALL PROBE GRADIOMETRY ] A novel Scanning Hall probe gradiometer has been developed and a new method to image x, y & z components of the